$$\begin{gathered} \mathrm{If} \mathrm{\alpha } + \mathrm{\beta } = - 2 \mathrm{and} {\mathrm{\alpha }}^3 + {\mathrm{\beta }}^3 = - 56, \mathrm{then} \\ \mathrm{the} \mathrm{quadratic} \mathrm{equation} \mathrm{whose} \mathrm{roots} \mathrm{are} \mathrm{\alpha } \mathrm{and} \mathrm{\beta } \end{gathered}$$

• x² + 2x - 16 = 0

• x² + 2x + 15 = 0

• x² + 2x - 12 = 0

• x² + 2x - 8 = 0

The cubic equation whose roots are thrice to each of the roots of x³ + 2x² - 4x + 1 = 0 is

• ${\mathrm{x}}^3 + 6{\mathrm{x}}^2 - 36\mathrm{x} + 27 = 0$

• ${\mathrm{x}}^3 + 6{\mathrm{x}}^2 + 36\mathrm{x} + 27 = 0$

• ${\mathrm{x}}^3 - 6{\mathrm{x}}^2 - 36\mathrm{x} + 27 = 0$

• ${\mathrm{x}}^3 - 6{\mathrm{x}}^2 + 36\mathrm{x} + 27 = 0$

The sum of the fourth powers of the roots of the equation

x³ + x + 1 = 0 is

• - 2

• - 1

• 1

• 2

let $\mathrm{\alpha } \mathrm{and} \mathrm{\beta }$ be the roots of the quadratic equation ax² + bx + c = 0. Observe the lists given below

	List-I		List-II
(i)	$\mathrm{\alpha } = \mathrm{\beta }$	(A)	(ac2)1/3 + (a2c)1/3 + b = 0
(ii)	$\mathrm{\alpha } = 2\mathrm{\beta }$	(B)	2b2 = 9ac
(iii)	$\mathrm{\alpha } = 3\mathrm{\beta }$	(C)	b2 = 6ac
(iv)	$\mathrm{\alpha } = {\mathrm{\beta }}^2$	(D)	3b2 = 16ac
		(E)	b2 = 4ac
		(F)	(ac2)1/3 + (a2c)1/3 = b

The correct match of List-I from List-II is

The roots (x - a) (x - a - 1) + (x - a - 1) (x - a - 2) + (x - a) (x - a - 2) = 0, a $\in$ R are always

• equal

• imaginary

• real and distinct

• rational and equal

Let f(x) = x + ax + b, where a, b $\in$ R. If f(x) = 0 has all-its roots imaginary, then the roots of f(x) + f'(x) + f"(x) = 0 are

• real and distinct

• imaginary

• equal

• rational and equal

If $\mathrm{\alpha }, \mathrm{\beta }, \mathrm{\gamma }$ are the roots of x³ + 4x + 1 = 0, then the equation whose roots are $\frac{{\mathrm{\alpha }}^3}{\mathrm{\beta } + \mathrm{\gamma }}, \frac{{\mathrm{\beta }}^2}{\mathrm{\gamma } + \mathrm{\alpha }}, \frac{{\mathrm{\gamma }}^2}{\mathrm{\alpha } + \mathrm{\beta }}$ is

• x³ - 4x - 1 = 0

• x³ - 4x + 1 = 0

• x³ + 4x - 1 = 0

• x³ + 4x + 1 = 0

If $\mathrm{\alpha } \mathrm{and} \mathrm{\beta }$ are the roots of x² - 2x + 4 = 0, then the value of ${\mathrm{\alpha }}^6 + {\mathrm{\beta }}^6$ is

• 32

• 64

• 128

• 256

If n is an integer which leaves remainder one when divided by three, then ${\left(1 + \sqrt{3}\mathrm{i}\right)}^{\mathrm{n}} + {\left(1 - \sqrt{3}\mathrm{i}\right)}^{\mathrm{n}}$ equals

• - 2^{n + 1}

• 2^{n + 1}

• - (- 2)ⁿ

• - 2ⁿ

240.

If ∝, ß, y are the roots of the equation x³ - 6x² + 11x - 6 = 0 and if a = ∝² + ß² + γ², b = ∝ß + ßγ + γ∝ and  c = (∝ + ß)(ß + γ)(γ + ∝), then the correct inequality among the following is

• $\mathrm{a} < \mathrm{b} < \mathrm{c}$

• $\mathrm{b} < \mathrm{a} < \mathrm{c}$

• $\mathrm{b} < \mathrm{c} < \mathrm{a}$

• $\mathrm{c} < \mathrm{a} < \mathrm{b}$